For a given irrational number $\theta$, we define Toeplitz operators with
symbols in the irrational rotation algebra ${\mathcal A}_\theta$,
and we show that the $C^*$-algebra $\mathcal T({\mathcal
A}_\theta)$ generated by these Toeplitz operators is an extension
of ${\mathcal A}_\theta$ by the algebra of compact operators. We
then use these extensions to explicitly exhibit generators of the
group $KK^1({\mathcal A}_\theta,\mathbb C)$. We also prove an
index theorem for $\mathcal T({\mathcal A}_\theta)$ that
generalizes the standard index theorem for Toeplitz operators on
the circle.